A232623 -id:A232623 - cp's OEIS frontend

A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 25, 35, 42, 56, 67, 89, 106, 138, 166, 211, 254, 321, 384, 479, 575, 709, 848, 1040, 1239, 1508, 1795, 2168, 2574, 3095, 3661, 4379, 5171, 6154, 7246, 8592, 10088, 11915, 13960, 16425, 19197, 22520, 26253, 30702, 35718

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

			a(1) = 1: [2].
a(2) = 2: [2,2], [2,1,1].
a(3) = 2: [2,2,2], [3,1,1,1].
a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].
a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].
a(6) = 5: [2,2,2,2,2,2], [2,2,2,1,1,1,1,1,1], [3,3,1,1,1,1,1,1], [4,2,1,1,1,1,1,1], [6,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A002865, A091602, A232623, A332051.

Partial sums give A133041.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
a:= n-> b(2*n, 1, n)-`if`(n=0, 0, b(2*n, 1, n-1)):
seq(a(n), n=0..60);

CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* Jean-François Alcover, Dec 18 2016 *)

G.f.: x/(1-x) + Product_{k>=2} 1/(1-x^k).
a(0) = 1, a(n) = 1 + A002865(n) = 1 + A000041(n)-A000041(n-1) for n>0.
a(n) = A091602(2n,n) = A096144(2n,n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - Vaclav Kotesovec, Oct 25 2018

A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36

Offset: 0

Henry Bottomley, Apr 20 2001

			T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
   1, 1, 2, 2, 3,  4,  5,  6,  8, ...
      2, 2, 4, 5,  7,  9, 13, 16, ...
         3, 4, 6,  9, 12, 16, 22, ...
            5, 6, 10, 13, 19, 25, ...
               7, 10, 14, 20, 27, ...
                  11, 14, 21, 28, ...
                      15, 21, 29, ...
                          22, 29, ...
                              30, ...

Alois P. Heinz, Columns k = 0..140, flattened

Rows effectively include A000007, A000009, A000726, A001935, A035959.
Main diagonal is A000041.
A061198 is the same table but includes cases where n>k.
T(n,2*n) gives: A232623.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(k$2, n):
seq(seq(T(n, k), n=0..k), k=0..12);  # Alois P. Heinz, Nov 27 2013

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k], {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A232605 Number of compositions of 2n into parts with multiplicity <= n.

Original entry on oeis.org

1, 1, 7, 26, 114, 459, 1892, 7660, 31081, 125464, 506025, 2036706, 8189555, 32894825, 132033140, 529614616, 2123365038, 8509634259, 34092146068, 136546197412, 546774790297, 2189060331762, 8762770476060, 35072837719356, 140363923730474, 561697985182654

Offset: 0

Alois P. Heinz, Nov 26 2013

nonn

a(n) = A243081(2n,n) = Sum_{i=0..n} A242447(2n,i).

			a(1) = 1: [2].
a(2) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
a(3) = 26: [6], [5,1], [4,2], [3,3], [2,4], [1,5], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [2,2,2], [1,3,2], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [2,1,2,1], [1,2,2,1], [1,1,3,1], [3,1,1,1], [2,2,1,1], [1,1,2,2], [1,1,1,3], [1,3,1,1], [2,1,1,2], [1,2,1,2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A232623, A232665.

a:= proc(n) option remember;
     `if`(n<5, [1, 1, 7, 26, 114][n+1],
      (2*(n-1)*(11092322562903*n^3 -66692687083623*n^2
       +117736395568913*n -51473509383358) *a(n-1)
      -(17386283060104*n^4 -178154697569624*n^3 +652039987731328*n^2
       -984836231488344*n +485931992440304) *a(n-2)
      -(89948343833304*n^4 -664733317200192*n^3 +1662507315916082*n^2
       -1594206267597886*n +485625773146800) *a(n-3)
      +(92866735410328*n^4 -1047423564207444*n^3 +4160804083968884*n^2
       -6634447008138888*n +3217864137236880) *a(n-4)
      -16*(n-5)*(2*n-9)*(310469340359*n^2 -847919784312*n
       +494768703748) *a(n-5)) / (5*n*(n-1)*
      (681426847222*n^2 -3587414825361*n +4663189129034)))
    end:
seq(a(n), n=0..40);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]];
A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n - 1)], b[n, n, 0, k]];
a[n_] := A[2 n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 31 2017, after Alois P. Heinz *)

Recurrence: 5*(n-2)*(n-1)*n*(1258*n^4 - 11230*n^3 + 37013*n^2 - 53645*n + 28764)*a(n) = 2*(n-2)*(n-1)*(17612*n^5 - 159736*n^4 + 538872*n^3 - 824111*n^2 + 541051*n - 107568)*a(n-1) - 4*(n-2)*(5032*n^5 - 44925*n^4 + 134332*n^3 - 137541*n^2 - 6614*n + 52596)*a(n-2) - 2*(83028*n^7 - 1074550*n^6 + 5758938*n^5 - 16516699*n^4 + 27297714*n^3 - 25934731*n^2 + 13070460*n - 2661120)*a(n-3) + 8*(n-4)*(n-1)*(2*n-7)*(1258*n^4 - 6198*n^3 + 10871*n^2 - 8277*n + 2160)*a(n-4). - Vaclav Kotesovec, Nov 27 2013

a(n) ~ 2^(2*n-1). - Vaclav Kotesovec, Nov 27 2013

A364245 Number of parts in all partitions of 2n into parts with multiplicity <= n.

Original entry on oeis.org

0, 1, 8, 24, 65, 150, 330, 657, 1274, 2338, 4172, 7203, 12171, 20045, 32474, 51623, 80867, 124841, 190406, 286857, 427758, 631367, 923544, 1339226, 1926798, 2751094, 3900931, 5494411, 7690923, 10701618, 14808183, 20380969, 27910066, 38035633, 51597166, 69685656

Offset: 0

Alois P. Heinz, Jul 15 2023

nonn

			a(2) = 8 = 3 + 2 + 2 + 1: [2,1,1], [2,2], [3,1], [4].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A210485, A232623.

a:= proc(k) option remember; local b; b:=
      proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
        add((l-> l+[0, l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, k))))
      end: b(2*k$2)[2]
    end:
seq(a(n), n=0..37);

a(n) = A210485(2n,n).

cp's OEIS Frontend

A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

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A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

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A232605 Number of compositions of 2n into parts with multiplicity <= n.

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A364245 Number of parts in all partitions of 2n into parts with multiplicity <= n.

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